User:Rjowsey/sandbox

From Wikipedia, the free encyclopedia

In mathematics, the complexification of a real vector space results in a complex vector space (over the complex number field). To "complexify" a space means that we extend ordinary multiplication by real numbers to include multiplication by complex numbers. In mathematical physics, when we complexify a real coordinate space Rn we create a complex coordinate space Cn, referred to in differential geometry as a "complex manifold".

The Minkowski space of Special and General Relativity (GR) is a "pseudo-euclidean" vector space. The spacetime underlying Einstein's Field Equations, which mathematically describe gravitation, is a 4-dimensional "Reimannian manifold", in which the four dimensions of space and time (x, y, z, ict) are regarded as spatial distances, measured in metres (or light-seconds), and fundamentally interchangeable. Any point in complex spacetime can be represented by a vector in the complex plane; its real position (x) denoted by cos(φ), and its imaginary time (ict) value by sin(φ).

In 1919, Theodor Kaluza posted his 5-dimensional extension of General Relativity to Einstein,[1] who was deeply impressed at the natural way in which the equations of electromagnetism emerged from Kaluza's five-dimensional maths. In 1926, Oskar Klein suggested[2] that Kaluza's extra dimension might be "curled up" into an infinitesimal circle, as if a circular topology is hidden within every point in space. Instead of being another spatial dimension, the extra dimension could be thought of as an angle, which created a hyper-dimension as it spun through 360°.

For several decades after publishing his General Theory of Relativity in 1915, Einstein tried to unify gravitational field theory with Maxwell's electromagnetism, to create a unified field theory which could explain the gravitational and electromagnetic fields in terms of a universal "gravito-electro-magnetic" field. He was attempting to consolidate the fundamental forces and the elementary particles into a single uniform field theory. His last few scientific papers reveal his valiant struggles to unify GR and QM in Kaluza's 5-dimensional spacetime.

"Scientifically, I am still lagging because of the same mathematical difficulties which make it impossible for me to affirm or contradict my more general relativistic field theory [...]. I will not be able to finish it [the work]; It will be forgotten and at a later time arguably mist be re-discovered. It happened this way with so many problems." — Albert Einstein, correspondence with Maurice Solovine, 25 Nov 1948[3]

In 1953, Wolfgang Pauli generalised[4] the Kaluza-Klein theory to a six-dimensional space, and (using dimensional reduction) derived the essentials of an SU(2) gauge theory (applied in QM to the electroweak interaction), as if Klein's "curled up" circle had become the surface of an infinitesimal hyper-sphere.

Although the wavefunctions and particles of quantum mechanics (QM) are thought of as inhabiting the exact-same Minkowski space as described by GR, the (configuration) state space in QM is actually a multi-dimensional, complex (Hilbert) vector space. Imaginary numbers fall from every equation, and behind all the mathematics[5] of QM is Euler's formula, "the most remarkable formula in mathematics" according to Richard Feynman:[6]

The exact-same equation defines Kaluza-Klein's extra "curled up" spatial dimension, and it is the basis of the complex numbers, and of the unitary group U(1), also known as the circle group, which is the most fundamental symmetry in the universe.

Deep Dimensional Analysis[edit]

History[edit]

During the late 1860's, James Clerk Maxwell's ideas about electromagnetism gradually became more mathematically complex. His spacetime geometry contained two imaginary dimensions to accommodate the electric potential E and the magnetic field H, plus another imaginary dimension for the gravitational potential V, all three being mathematically orthogonal to real Euclidean space. In his discussion of the findings of the great electromagnetic experimentalist Michael Faraday, these comprised six spatial dimensions (three real and three imaginary).

"I am getting converted to Quaternions, and have put some in my book, in a heretical form..." — James Clerk Maxwell, correspondence with Prof. Lewis Campbell, 19 Oct 1872[7]

In his scientific description of electromagnetism, Maxwell used what he called a "heretical form" of quaternion algebra, which explicitly separated the three imaginary dimensions from the real part. He stated emphatically that tensors and vectors were inadequate mathematical tools to correctly encapsulate the electromagnetic fields and forces. He also quietly discussed with colleagues how one might detect and measure "non-observable" or "hidden" spatial dimensions, which he conceived of as "storing energy", both kinetic and potential, in the elastic fabric of space itself.[8]

"The peculiarity of our space is that of its three dimensions, none is before or after another. As is x, so is y, and so is z. If you have 4 dimensions, this becomes a puzzle. For first, if three of them are in our space, then which three? Also, if we lived in space of m dimensions, but were only capable of thinking n of them, then first, which n? Second, if so, things would happen requiring the rest to explain them, and so we should either be stultified or made wiser. I am quite sure that the kind of continuity which has four dimensions all co-equal, is not to be discovered by merely generalising Cartesian space equations." — James Clerk Maxwell, in correspondence with C.J. Monro, Esq., 15 Mar 1871[9]
James Clerk Maxwell
James Clerk Maxwell (1831–1879)

His preferred quaternion notation was eliminated from A Treatise on Electricity and Magnetism[10] at the insistence of his publisher (over his strenuous objections), because very few scientists at the time could understood the maths. Maxwell regrettably passed away in 1879 at age 48, when he was only partway through his revision for the second edition.

Maxwell played a major role in establishing the modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived.[11] Although he defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of Newton's law of universal gravitation in which the gravitational constant G is taken as unity, giving M = L3T−2.[12]

By assuming a form of Coulomb's law in which Coulomb's constant ke is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were Q = L3/2M1/2T−1,[13] which, after substituting his M = L3T−2 equation for mass, results in charge having the same fundamental dimensions as mass, viz. Q = L3T−2. This peculiarity must have intrigued and puzzled him, but he apparently never discussed it in his lectures, correspondence or scientific writings.

The fundamental physical constants[edit]

The primary physical constant is the speed of light in vacuum (c0), which has unitary space-time dimensions of L/T, i.e. metres per second.

Another fundamental is Newton's gravitational constant (G), with SI dimensions of L3/T2M. When Maxwell's dimensions for mass [12] (M = L3/T2) are substituted into the SI dimensions, the gravitational constant is shown to be dimensionless in elemental space-time, viz. L0/T0.

Planck's constant (h), the fundamental ratio of a quantum of energy to its wavefunction's frequency (T−1), has SI dimensions of L2M/T. Substituting Maxwell's dimensions for mass (M=L3/T2) shows that the Planck quantum of Action has fundamental dimensions of L5/T3.

Maxwell determined[12] that the unit of elementary charge (e) has dimensions of (L3M/T2)½. Substituting his mass dimensions (M = L3/T2) reveals that charge has the fundamental dimensions of (L6/T4)1/2, i.e. Q = L3/T2.

The Boltzmann constant (kB) is defined as the energy in Joules per degree of temperature (Θ), having SI dimensions of L2M/T2Θ. Substituting M = L3/T2 reveals the Boltzmann constant to have space-time dimensions of L5/T4 (energy) per degree K.

Universal physical constants normalized in 2 dimensions (L,T)
Constant Symbol SI Dimensions L/T Dimensions
Speed of light in vacuum c0 LT−1 LT−1
Gravitational constant G L3M−1 T−2 L0T0
Planck constant h L2MT−1 L5T−3
Coulomb constant ke L3MT−2Q−2 L0T0
Boltzmann constant kB L2MT−2Θ−1 L5T−4Θ−1

The Planck units[edit]

The Planck Units are "natural units" of measurement defined exclusively in terms of five universal physical constants, viz. c, G, ħ, ke and kB, such that these constants have the numerical value of 1 when expressed in terms of the Planck units.

The base spatial unit is the Planck length (ℓP), defined as the distance traveled by light in vacuum during one Planck time (tP). The numerical value of ℓP is calculated from (ħGc−3)1/2, the fundamental space-time dimensions of which resolve as (L5T−3⋅L−3T3)1/2 = L.

The five base Planck units, viz. length, time, mass, charge and temperature, have traditionally been dimensioned in terms of the base SI units L, T, M, Q and Θ. However, Maxwell's factoring[12] of mass and charge into the more fundamental space-time dimensions of L3T−2 permits a deep two-dimensional analysis of the base and derived Planck units.

Since the gravitational constant G and the vacuum permittivity (electric constant) ε0 are dimensionless in L/T space-time basis units, they can be factored out of the Planck units, thereby simplifying the dimensional analysis. For example, Planck area is defined as ħG/c3, which simplifies to L5T−3 ⋅ L−3T3 = L2. Similarly, Planck current is defined as (4πε0c6/G)1/2, which resolves to (L6T−6)1/2 = L3T−3.

Thus, the fundamental space-time dimensions for each of the Planck units can be derived from their defining expressions. However, it is considerably easier to simply substitute L3T−2 for M and Q in the conventional SI dimensions of the Planck quantities, as follows:

The Planck units normalized in 2 dimensions (L,T)
Name SI Dimensions L/T Dimensions
Planck length L L
Planck time T T
Planck area L2 L2
Planck volume L3 L3
Planck mass M L3T−2
Planck charge Q L3T−2
Planck momentum LMT−1 L4T−3
Planck force LMT−2 L4T−4
Planck energy L2MT−2 L5T−4
Planck power L2MT−3 L5T−5
Planck density ML−3 T−2
Planck intensity MT−3 L3T−5
Planck frequency T−1 T−1
Planck pressure L−1MT−2 L2T−4
Planck current QT−1 L3T−3
Planck voltage L2MT−2Q−1 L2T−2
Planck resistance L2MT−1Q−2 L−1T

To facilitate further analysis, these quantities can be arranged into a log-log space/time matrix, whose columns represent incrementing powers of Planck length (Ln) and whose rows represent increasing powers of inverse-time (Tm):

The Planck units arranged in a 6D space/time matrix
L/T L0 L1 L2 L3 L4 L5
T0 0
Constants

Length

Surface Area

Volume

4D-Volume

5D-Volume
T−1
Frequency

Velocity

Magnetic flux

Flow rate
T−2
Mass density

Acceleration

Voltage

Mass, Charge
T−3
Current flux

Current

Momentum

Planck Action
T−4
Pressure

Force

Energy
T−5
Intensity

Power

Five mutually-orthogonal spatial dimensions are required to accommodate all the Planck units, notably the "higher dimensional" (L4, L5) quantities of momentum, force, action, energy and power. Three of the spatial dimensions are the real linear dimensions of Euclidean x,y,z space, viz. length, breadth and height. Like the "time dimension" of special relativity, defined by Einstein (1905) as √-1∙c∙t, the two extra spatial dimensions are mathematically imaginary by virtue of their orthogonality, i.e. being Wick-rotated relative to all the other dimensions. The space/time matrix has internal symmetries corresponding to the SU(3) × SU(2) × U(1) unitary group, consistent with the Standard Model.

In 2006, Paul Wesson determined[14] that an extra spatial coordinate x4 could be identified as ℓ = Gm/c2, which he termed the "Einstein gauge". Formulated in terms of momentum, i.e. ℓ = Gp/c3, this gauge corresponds to the L4 spatial dimension of the space/time matrix, from which emerges momentum, force, and pressure. Wesson also identified another spatial coordinate as ℓ = ħ/mc (dimensionally identical to ℓ = ħ/qc), which he termed the "Planck gauge". This ħ/qc gauge corresponds to the L5 spatial coordinate in the space/time matrix. The physical quantities of action, energy, power and intensity emerge from this imaginary dimension.

Electromagnetism[edit]

Substitution of Maxwell's L3T−2 for mass (M) and charge (Q) in the SI dimensions of the electromagnetic (EM) quantities effects a "flattening" of their dimensionality to just spatial length and time, as follows:

The electromagnetic units in 2 space/time dimensions (L, T)
Quantity Symbol Unit SI Dimensions L/T Dimensions
Electric Charge Q coulomb Q L3T−2
Electric Current I ampere QT−1 L3T−3
Electric Potential Δ volt L2MT−2Q−1 L2T−2
Magnetic Field Strength H ampere per metre QL−1T−1 L2T−3
Electric Displacement D coulomb per metre2 QL−2 LT−2
Electric Field Strength E volt per metre LMT−2Q−1 LT−2
Current Density J ampere per metre2 QT−1L−2 LT−3
Charge Density ρq coulomb per metre3 QL−3 T−2
Electric Resistance R ohm L2MT−1Q−2 L−1T
Capacitance C farad T2Q2L−2M−1 L
Permittivity ε farad per metre T2Q2L−3M−1 1
Magnetic Flux Density B tesla MT−1Q−1 T−1
Magnetic Potential A weber per metre LMT−1Q−1 LT−1
Magnetic Flux Φ weber L2MT−1Q−1 L2T−1
Energy Density ρe joule per metre3 ML−1T−2 L2T−4
Inductance L henry ML2Q−2 L−1T2
Permeability μ henry per metre MLQ−2 L−2T2
Magnetic Momentum p newton-second LMT−1 L4T−3
Electromagnetic Force F newton LMT−2 L4T−4
Electromagnetic Action S joule-second L2MT−1 L5T−3
Electric Potential Energy U joule L2MT−2 L5T−4
Radiant EM Energy E joule L2MT−2 L5T−4
Poynting Vector S watt per metre2 MT−3 L3T−5
Power P watt L2MT−3 L5T−5

As with the Planck Units, these quantities can be arranged in a 6-dimensional space/time matrix, with columns representing powers of Planck length (Ln), and rows which represent powers of inverse-time (Tm). Noting that the differential (or gradient) of any quantity with respect to time (d/dt) is the unit immediately below it, and that the differential of a quantity with respect to space (d/ds or ∇) is the unit to its left, Maxwell's equations can be discerned in the relationships between these electromagnetic quantities. Within the log-log matrix, multiplication is effected by adding the two units' dimensional indices, e.g. LaTb × LcTd = L(a+c)T(b+d), and division is performed by subtracting the denominator's space/time indices from the numerator's.

The electromagnetic quantities arranged in a 6D space/time matrix
L/T L0 L1 L2 L3 L4 L5
T0 Electric constant
0 = 1
Wavelength λ
Capacitance C
Surface area Volume 4D-volume 5D-volume
T−1 Magnetic flux
density B
Magnetic vector
potential A
Magnetic flux
Φ
∫ Φ⋅ds
∫ Q dt
T−2 Charge density
ρq = ∇ ⋅ E
Electric field
strength E (D)
Electrical potential
Δ
Charge
Q
Q⋅ds
p dt
S⋅dt
T−3 Current density
J = ∇ × H
Magnetic field
strength H
Current
I
Momentum
p = QA
Action
S
T−4 Energy density
ρe = BH
F = A × H Force
F = Q (E + v x B)
Energy
Uε = Q⋅Δ
T−5 2S ∇ ⋅ S Poynting vector
S = E × H
dF/dt = ∇P Power
P = ΔI

Imaginary Dimensions[edit]

Imaginary spatial dimensions in complex 6D space/time may be formulated using various equivalent expressions, all of which resolve to the dimension of spatial distance (L).

The "Einstein gauge" is canonically formulated[14] as Gm/c2 (half the Schwarzschild radius), but is more usefully expressed in terms of momentum, viz. ℓ = p/c3, particularly in the context of 6-dimensional Special Relativity. It can also be expressed in terms of kinetic energy as ℓ= Ek/c4. In the context of electromagnetism, this metric is best formulated in terms of magnetic momentum divided by current (flow of charge): ℓ = p/I. In quantum mechanics, this imaginary dimension can be formulated in terms of a wavefunction's frequency of oscillation: ℓ = ƒ/c4 (where frequency ƒ = c/λ and energy E = ħƒ).

The "Planck gauge" was originally formulated[14] as ħ/mc in the context of higher-dimensional (Kaluza–Klein) gravitation. In electromagnetism, this imaginary dimension may be equivalently expressed as ℓ = ħ/qc = q/c2. In regards to electric potential (voltage), it is formulated by ℓ = q/V = ħc/qV, and in quantum mechanics it is most usefully expressed in terms of potential energy, viz. ℓ = iħc/E0 = iq2/E0 (where E0 = mc2 = qV).

The 5th spatial dimension is associated with potential energy and inverse-time, analogous to the 4th imaginary dimension's association with spatial position and momentum (or kinetic energy). The complex dimensions associated with potential and kinetic energy are clearly orthogonal, since total energy squared is given by E2 = E02 + Ek2. Heisenburg's Uncertainty Principle can also be expressed in terms of position and momentum, as σxσpħ/2, or in terms of potential energy and time, viz. σuσtħ/2.

Special Relativity[edit]

The animated graphic below simulates a test particle with Planck mass being accelerated to the speed of light, where it has Planck momentum and Planck kinetic energy. The simplified Minkowski diagram at the top-left shows the "4D Lorentz rotation" of the moving frame of reference inhabited by the particle. The bottom-left L4 projection shows length contraction in the moving frame of reference, with the real spatial dimension (ʀ) plotted against the "Einstein gauge" representing imaginary momentum, having the canonical unit of Gp/c3 in hyperspatial Special Relativity. The top-right L5 projection shows time dilation approaching infinity at light speed; imaginary time ict is plotted against the "Planck gauge", the imaginary x-axis unit canonically defined by λᵩ = ħ/mc, the Compton wavelength.

Special Relativity in 6 dimensions<\p>

The Lorentz factor is the inverse-cosine of the phase angle (0 < φ < π/2), i.e. γ = 1/cos(φ), and the ratio of the particle's velocity to light speed is β = v/c = sin(φ). Thus, time dilation and length contraction simplify to τᵩ = t∙cos(φ) and ʀᵩ = r∙cos(φ). The y-axis in the bottom-right projection represents the imaginary component of the particle's kinetic energy, while the x-axis represents the imaginary component of its potential energy (mc2) in units of Planck energy (EP). The particle's total energy Eᵩ is a function of √((t∙sin(φ))2 + (r∙sin(φ))2). At light speed, the particle's rest-mass energy and its momentum can be seen as inhabiting two imaginary spatial dimensions. A Planck mass moving at the velocity of light has total energy of √2∙EP (assuming there is no energy loss due to gravitational radiation). The particle's matter-wave has a de Broglie wavelength λᵩ of one Planck length, and its wavefunction is oscillating at Planck frequency.

See also[edit]

References[edit]

  1. ^ Pais, Abraham (1982). Subtle is the Lord ...: The Science and the Life of Albert Einstein. Oxford: Oxford University Press. pp. 329–330.
  2. ^ Oskar Klein (1926). "Quantentheorie und fünfdimensionale Relativitätstheorie". Zeitschrift für Physik A. 37 (12): 895–906. Bibcode:1926ZPhy...37..895K. doi:10.1007/BF01397481.
  3. ^ Hubert F. M. Goenner (2014). "On the History of Unified Field Theories Part II (ca. 1930 — ca. 1965)". Living Reviews in Relativity. 17 (5): 75.
  4. ^ N. Straumann (2000). "On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
  5. ^ Mark Davidson (2011). "The Lorentz-Dirac equation in complex space-time" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
  6. ^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. p. 22-10. ISBN 0-201-02010-6.
  7. ^ Lewis Campbell, William Garnett (2010), The Life of James Clerk Maxwell: With a Selection from His Correspondence and Occasional Writings and a Sketch of His Contributions to Science, Cambridge University Press, p. 383 ]
  8. ^ Lewis Campbell, William Garnett (2010), The Life of James Clerk Maxwell: With a Selection from His Correspondence and Occasional Writings and a Sketch of His Contributions to Science, Cambridge University Press, p. 550 ]
  9. ^ Lewis Campbell, William Garnett (2010), The Life of James Clerk Maxwell: With a Selection from His Correspondence and Occasional Writings and a Sketch of His Contributions to Science, Cambridge University Press, p. 380 ]
  10. ^ Maxwell, James Clerk (1873), A Treatise on Electricity and Magnetism
  11. ^ Roche, John J (1998), The Mathematics of Measurement: A Critical History, London: Springer, p. 203, ISBN 978-0-387-91581-4, [1] {{citation}}: External link in |quote= (help)
  12. ^ a b c d Maxwell, James Clerk (1873), A Treatise on Electricity and Magnetism, p. 4
  13. ^ Maxwell, James Clerk (1873), A Treatise on Electricity and Magnetism, p. 45
  14. ^ a b c Wesson, Paul S. (2006), Five-dimensional physics: classical and quantum consequences of Kaluza-Klein cosmology, Singapore: World Scientific, p. 103, ISBN 981-256-661-9

Selected papers[edit]

Further reading[edit]

External links[edit]

Category:Physics
Category:Spacetime
Category:Dimensional analysis